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 "cells": [
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   "source": [
    "Installing (updating) the following libraries for your Sagemaker\n",
    "instance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eeba8632",
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   "source": [
    "!pip install .. # installing d2l\n"
   ]
  },
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    "# 概率\n",
    ":label:`sec_prob`\n",
    "\n",
    "简单地说，机器学习就是做出预测。\n",
    "\n",
    "根据病人的临床病史，我们可能想预测他们在下一年心脏病发作的*概率*。\n",
    "在飞机喷气发动机的异常检测中，我们想要评估一组发动机读数为正常运行情况的概率有多大。\n",
    "在强化学习中，我们希望智能体（agent）能在一个环境中智能地行动。\n",
    "这意味着我们需要考虑在每种可行的行为下获得高奖励的概率。\n",
    "当我们建立推荐系统时，我们也需要考虑概率。\n",
    "例如，假设我们为一家大型在线书店工作，我们可能希望估计某些用户购买特定图书的概率。\n",
    "为此，我们需要使用概率学。\n",
    "有完整的课程、专业、论文、职业、甚至院系，都致力于概率学的工作。\n",
    "所以很自然地，我们在这部分的目标不是教授整个科目。\n",
    "相反，我们希望教给读者基础的概率知识，使读者能够开始构建第一个深度学习模型，\n",
    "以便读者可以开始自己探索它。\n",
    "\n",
    "现在让我们更认真地考虑第一个例子：根据照片区分猫和狗。\n",
    "这听起来可能很简单，但对于机器却可能是一个艰巨的挑战。\n",
    "首先，问题的难度可能取决于图像的分辨率。\n",
    "\n",
    "![不同分辨率的图像 ($10 \\times 10$, $20 \\times 20$, $40 \\times 40$, $80 \\times 80$, 和 $160 \\times 160$ pixels)](../img/cat-dog-pixels.png)\n",
    ":width:`300px`\n",
    ":label:`fig_cat_dog`\n",
    "\n",
    "如 :numref:`fig_cat_dog`所示，虽然人类很容易以$160 \\times 160$像素的分辨率识别猫和狗，\n",
    "但它在$40\\times40$像素上变得具有挑战性，而且在$10 \\times 10$像素下几乎是不可能的。\n",
    "换句话说，我们在很远的距离（从而降低分辨率）区分猫和狗的能力可能会变为猜测。\n",
    "概率给了我们一种正式的途径来说明我们的确定性水平。\n",
    "如果我们完全肯定图像是一只猫，我们说标签$y$是\"猫\"的*概率*，表示为$P(y=$\"猫\"$)$等于$1$。\n",
    "如果我们没有证据表明$y=$“猫”或$y=$“狗”，那么我们可以说这两种可能性是相等的，\n",
    "即$P(y=$\"猫\"$)=P(y=$\"狗\"$)=0.5$。\n",
    "如果我们不十分确定图像描绘的是一只猫，我们可以将概率赋值为$0.5<P(y=$\"猫\"$)<1$。\n",
    "\n",
    "现在考虑第二个例子：给出一些天气监测数据，我们想预测明天北京下雨的概率。\n",
    "如果是夏天，下雨的概率是0.5。\n",
    "\n",
    "在这两种情况下，我们都不确定结果，但这两种情况之间有一个关键区别。\n",
    "在第一种情况中，图像实际上是狗或猫二选一。\n",
    "在第二种情况下，结果实际上是一个随机的事件。\n",
    "因此，概率是一种灵活的语言，用于说明我们的确定程度，并且它可以有效地应用于广泛的领域中。\n",
    "\n",
    "## 基本概率论\n",
    "\n",
    "假设我们掷骰子，想知道看到1的几率有多大，而不是看到另一个数字。\n",
    "如果骰子是公平的，那么所有六个结果$\\{1, \\ldots, 6\\}$都有相同的可能发生，\n",
    "因此我们可以说$1$发生的概率为$\\frac{1}{6}$。\n",
    "\n",
    "然而现实生活中，对于我们从工厂收到的真实骰子，我们需要检查它是否有瑕疵。\n",
    "检查骰子的唯一方法是多次投掷并记录结果。\n",
    "对于每个骰子，我们将观察到$\\{1, \\ldots, 6\\}$中的一个值。\n",
    "对于每个值，一种自然的方法是将它出现的次数除以投掷的总次数，\n",
    "即此*事件*（event）概率的*估计值*。\n",
    "*大数定律*（law of large numbers）告诉我们：\n",
    "随着投掷次数的增加，这个估计值会越来越接近真实的潜在概率。\n",
    "让我们用代码试一试！\n",
    "\n",
    "首先，我们导入必要的软件包。\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "fffe28f9",
   "metadata": {
    "execution": {
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     "pytorch"
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   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "from torch.distributions import multinomial\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8bfc741",
   "metadata": {
    "origin_pos": 5
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   "source": [
    "在统计学中，我们把从概率分布中抽取样本的过程称为*抽样*（sampling）。\n",
    "笼统来说，可以把*分布*（distribution）看作对事件的概率分配，\n",
    "稍后我们将给出的更正式定义。\n",
    "将概率分配给一些离散选择的分布称为*多项分布*（multinomial distribution）。\n",
    "\n",
    "为了抽取一个样本，即掷骰子，我们只需传入一个概率向量。\n",
    "输出是另一个相同长度的向量：它在索引$i$处的值是采样结果中$i$出现的次数。\n"
   ]
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    "origin_pos": 7,
    "tab": [
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    {
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      "text/plain": [
       "tensor([0., 0., 1., 0., 0., 0.])"
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     "execution_count": 2,
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   "source": [
    "fair_probs = torch.ones([6]) / 6\n",
    "multinomial.Multinomial(1, fair_probs).sample()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "076723aa",
   "metadata": {
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   "source": [
    "在估计一个骰子的公平性时，我们希望从同一分布中生成多个样本。\n",
    "如果用Python的for循环来完成这个任务，速度会慢得惊人。\n",
    "因此我们使用深度学习框架的函数同时抽取多个样本，得到我们想要的任意形状的独立样本数组。\n"
   ]
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  {
   "cell_type": "code",
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   "id": "aef6c2b1",
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    "origin_pos": 12,
    "tab": [
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    {
     "data": {
      "text/plain": [
       "tensor([5., 3., 2., 0., 0., 0.])"
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     "execution_count": 3,
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   "source": [
    "multinomial.Multinomial(10, fair_probs).sample()"
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   "cell_type": "markdown",
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   "metadata": {
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   "source": [
    "现在我们知道如何对骰子进行采样，我们可以模拟1000次投掷。\n",
    "然后，我们可以统计1000次投掷后，每个数字被投中了多少次。\n",
    "具体来说，我们计算相对频率，以作为真实概率的估计。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "22aadbd3",
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0.1550, 0.1820, 0.1770, 0.1710, 0.1600, 0.1550])"
      ]
     },
     "execution_count": 4,
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   "source": [
    "# 将结果存储为32位浮点数以进行除法\n",
    "counts = multinomial.Multinomial(1000, fair_probs).sample()\n",
    "counts / 1000  # 相对频率作为估计值"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af0608f2",
   "metadata": {
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   "source": [
    "因为我们是从一个公平的骰子中生成的数据，我们知道每个结果都有真实的概率$\\frac{1}{6}$，\n",
    "大约是$0.167$，所以上面输出的估计值看起来不错。\n",
    "\n",
    "我们也可以看到这些概率如何随着时间的推移收敛到真实概率。\n",
    "让我们进行500组实验，每组抽取10个样本。\n"
   ]
  },
  {
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    "counts = multinomial.Multinomial(10, fair_probs).sample((500,))\n",
    "cum_counts = counts.cumsum(dim=0)\n",
    "estimates = cum_counts / cum_counts.sum(dim=1, keepdims=True)\n",
    "\n",
    "d2l.set_figsize((6, 4.5))\n",
    "for i in range(6):\n",
    "    d2l.plt.plot(estimates[:, i].numpy(),\n",
    "                 label=(\"P(die=\" + str(i + 1) + \")\"))\n",
    "d2l.plt.axhline(y=0.167, color='black', linestyle='dashed')\n",
    "d2l.plt.gca().set_xlabel('Groups of experiments')\n",
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   "source": [
    "每条实线对应于骰子的6个值中的一个，并给出骰子在每组实验后出现值的估计概率。\n",
    "当我们通过更多的实验获得更多的数据时，这$6$条实体曲线向真实概率收敛。\n",
    "\n",
    "### 概率论公理\n",
    "\n",
    "在处理骰子掷出时，我们将集合$\\mathcal{S} = \\{1, 2, 3, 4, 5, 6\\}$\n",
    "称为*样本空间*（sample space）或*结果空间*（outcome space），\n",
    "其中每个元素都是*结果*（outcome）。\n",
    "*事件*（event）是一组给定样本空间的随机结果。\n",
    "例如，“看到$5$”（$\\{5\\}$）和“看到奇数”（$\\{1, 3, 5\\}$）都是掷出骰子的有效事件。\n",
    "注意，如果一个随机实验的结果在$\\mathcal{A}$中，则事件$\\mathcal{A}$已经发生。\n",
    "也就是说，如果投掷出$3$点，因为$3 \\in \\{1, 3, 5\\}$，我们可以说，“看到奇数”的事件发生了。\n",
    "\n",
    "*概率*（probability）可以被认为是将集合映射到真实值的函数。\n",
    "在给定的样本空间$\\mathcal{S}$中，事件$\\mathcal{A}$的概率，\n",
    "表示为$P(\\mathcal{A})$，满足以下属性：\n",
    "\n",
    "* 对于任意事件$\\mathcal{A}$，其概率从不会是负数，即$P(\\mathcal{A}) \\geq 0$；\n",
    "* 整个样本空间的概率为$1$，即$P(\\mathcal{S}) = 1$；\n",
    "* 对于*互斥*（mutually exclusive）事件（对于所有$i \\neq j$都有$\\mathcal{A}_i \\cap \\mathcal{A}_j = \\emptyset$）的任意一个可数序列$\\mathcal{A}_1, \\mathcal{A}_2, \\ldots$，序列中任意一个事件发生的概率等于它们各自发生的概率之和，即$P(\\bigcup_{i=1}^{\\infty} \\mathcal{A}_i) = \\sum_{i=1}^{\\infty} P(\\mathcal{A}_i)$。\n",
    "\n",
    "以上也是概率论的公理，由科尔莫戈罗夫于1933年提出。\n",
    "有了这个公理系统，我们可以避免任何关于随机性的哲学争论；\n",
    "相反，我们可以用数学语言严格地推理。\n",
    "例如，假设事件$\\mathcal{A}_1$为整个样本空间，\n",
    "且当所有$i > 1$时的$\\mathcal{A}_i = \\emptyset$，\n",
    "那么我们可以证明$P(\\emptyset) = 0$，即不可能发生事件的概率是$0$。\n",
    "\n",
    "### 随机变量\n",
    "\n",
    "在我们掷骰子的随机实验中，我们引入了*随机变量*（random variable）的概念。\n",
    "随机变量几乎可以是任何数量，并且它可以在随机实验的一组可能性中取一个值。\n",
    "考虑一个随机变量$X$，其值在掷骰子的样本空间$\\mathcal{S}=\\{1,2,3,4,5,6\\}$中。\n",
    "我们可以将事件“看到一个$5$”表示为$\\{X=5\\}$或$X=5$，\n",
    "其概率表示为$P(\\{X=5\\})$或$P(X=5)$。\n",
    "通过$P(X=a)$，我们区分了随机变量$X$和$X$可以采取的值（例如$a$）。\n",
    "然而，这可能会导致繁琐的表示。\n",
    "为了简化符号，一方面，我们可以将$P(X)$表示为随机变量$X$上的*分布*（distribution）：\n",
    "分布告诉我们$X$获得某一值的概率。\n",
    "另一方面，我们可以简单用$P(a)$表示随机变量取值$a$的概率。\n",
    "由于概率论中的事件是来自样本空间的一组结果，因此我们可以为随机变量指定值的可取范围。\n",
    "例如，$P(1 \\leq X \\leq 3)$表示事件$\\{1 \\leq X \\leq 3\\}$，\n",
    "即$\\{X = 1, 2, \\text{or}, 3\\}$的概率。\n",
    "等价地，$P(1 \\leq X \\leq 3)$表示随机变量$X$从$\\{1, 2, 3\\}$中取值的概率。\n",
    "\n",
    "请注意，*离散*（discrete）随机变量（如骰子的每一面）\n",
    "和*连续*（continuous）随机变量（如人的体重和身高）之间存在微妙的区别。\n",
    "现实生活中，测量两个人是否具有完全相同的身高没有太大意义。\n",
    "如果我们进行足够精确的测量，最终会发现这个星球上没有两个人具有完全相同的身高。\n",
    "在这种情况下，询问某人的身高是否落入给定的区间，比如是否在1.79米和1.81米之间更有意义。\n",
    "在这些情况下，我们将这个看到某个数值的可能性量化为*密度*（density）。\n",
    "高度恰好为1.80米的概率为0，但密度不是0。\n",
    "在任何两个不同高度之间的区间，我们都有非零的概率。\n",
    "在本节的其余部分中，我们将考虑离散空间中的概率。\n",
    "连续随机变量的概率可以参考深度学习数学附录中[随机变量](https://d2l.ai/chapter_appendix-mathematics-for-deep-learning/random-variables.html)\n",
    "的一节。\n",
    "\n",
    "## 处理多个随机变量\n",
    "\n",
    "很多时候，我们会考虑多个随机变量。\n",
    "比如，我们可能需要对疾病和症状之间的关系进行建模。\n",
    "给定一个疾病和一个症状，比如“流感”和“咳嗽”，以某个概率存在或不存在于某个患者身上。\n",
    "我们需要估计这些概率以及概率之间的关系，以便我们可以运用我们的推断来实现更好的医疗服务。\n",
    "\n",
    "再举一个更复杂的例子：图像包含数百万像素，因此有数百万个随机变量。\n",
    "在许多情况下，图像会附带一个*标签*（label），标识图像中的对象。\n",
    "我们也可以将标签视为一个随机变量。\n",
    "我们甚至可以将所有元数据视为随机变量，例如位置、时间、光圈、焦距、ISO、对焦距离和相机类型。\n",
    "所有这些都是联合发生的随机变量。\n",
    "当我们处理多个随机变量时，会有若干个变量是我们感兴趣的。\n",
    "\n",
    "### 联合概率\n",
    "\n",
    "第一个被称为*联合概率*（joint probability）$P(A=a,B=b)$。\n",
    "给定任意值$a$和$b$，联合概率可以回答：$A=a$和$B=b$同时满足的概率是多少？\n",
    "请注意，对于任何$a$和$b$的取值，$P(A = a, B=b) \\leq P(A=a)$。\n",
    "这点是确定的，因为要同时发生$A=a$和$B=b$，$A=a$就必须发生，$B=b$也必须发生（反之亦然）。因此，$A=a$和$B=b$同时发生的可能性不大于$A=a$或是$B=b$单独发生的可能性。\n",
    "\n",
    "### 条件概率\n",
    "\n",
    "联合概率的不等式带给我们一个有趣的比率：\n",
    "$0 \\leq \\frac{P(A=a, B=b)}{P(A=a)} \\leq 1$。\n",
    "我们称这个比率为*条件概率*（conditional probability），\n",
    "并用$P(B=b \\mid A=a)$表示它：它是$B=b$的概率，前提是$A=a$已发生。\n",
    "\n",
    "### 贝叶斯定理\n",
    "\n",
    "使用条件概率的定义，我们可以得出统计学中最有用的方程之一：\n",
    "*Bayes定理*（Bayes' theorem）。\n",
    "根据*乘法法则*（multiplication rule ）可得到$P(A, B) = P(B \\mid A) P(A)$。\n",
    "根据对称性，可得到$P(A, B) = P(A \\mid B) P(B)$。\n",
    "假设$P(B)>0$，求解其中一个条件变量，我们得到\n",
    "\n",
    "$$P(A \\mid B) = \\frac{P(B \\mid A) P(A)}{P(B)}.$$\n",
    "\n",
    "请注意，这里我们使用紧凑的表示法：\n",
    "其中$P(A, B)$是一个*联合分布*（joint distribution），\n",
    "$P(A \\mid B)$是一个*条件分布*（conditional distribution）。\n",
    "这种分布可以在给定值$A = a, B=b$上进行求值。\n",
    "\n",
    "### 边际化\n",
    "\n",
    "为了能进行事件概率求和，我们需要*求和法则*（sum rule），\n",
    "即$B$的概率相当于计算$A$的所有可能选择，并将所有选择的联合概率聚合在一起：\n",
    "\n",
    "$$P(B) = \\sum_{A} P(A, B),$$\n",
    "\n",
    "这也称为*边际化*（marginalization）。\n",
    "边际化结果的概率或分布称为*边际概率*（marginal probability）\n",
    "或*边际分布*（marginal distribution）。\n",
    "\n",
    "### 独立性\n",
    "\n",
    "另一个有用属性是*依赖*（dependence）与*独立*（independence）。\n",
    "如果两个随机变量$A$和$B$是独立的，意味着事件$A$的发生跟$B$事件的发生无关。\n",
    "在这种情况下，统计学家通常将这一点表述为$A \\perp  B$。\n",
    "根据贝叶斯定理，马上就能同样得到$P(A \\mid B) = P(A)$。\n",
    "在所有其他情况下，我们称$A$和$B$依赖。\n",
    "比如，两次连续抛出一个骰子的事件是相互独立的。\n",
    "相比之下，灯开关的位置和房间的亮度并不是（因为可能存在灯泡坏掉、电源故障，或者开关故障）。\n",
    "\n",
    "由于$P(A \\mid B) = \\frac{P(A, B)}{P(B)} = P(A)$等价于$P(A, B) = P(A)P(B)$，\n",
    "因此两个随机变量是独立的，当且仅当两个随机变量的联合分布是其各自分布的乘积。\n",
    "同样地，给定另一个随机变量$C$时，两个随机变量$A$和$B$是*条件独立的*（conditionally independent），\n",
    "当且仅当$P(A, B \\mid C) = P(A \\mid C)P(B \\mid C)$。\n",
    "这个情况表示为$A \\perp B \\mid C$。\n",
    "\n",
    "### 应用\n",
    ":label:`subsec_probability_hiv_app`\n",
    "\n",
    "我们实战演练一下！\n",
    "假设一个医生对患者进行艾滋病病毒（HIV）测试。\n",
    "这个测试是相当准确的，如果患者健康但测试显示他患病，这个概率只有1%；\n",
    "如果患者真正感染HIV，它永远不会检测不出。\n",
    "我们使用$D_1$来表示诊断结果（如果阳性，则为$1$，如果阴性，则为$0$），\n",
    "$H$来表示感染艾滋病病毒的状态（如果阳性，则为$1$，如果阴性，则为$0$）。\n",
    "在 :numref:`conditional_prob_D1`中列出了这样的条件概率。\n",
    "\n",
    ":条件概率为$P(D_1 \\mid H)$\n",
    "\n",
    "| 条件概率 | $H=1$ | $H=0$ |\n",
    "|---|---|---|\n",
    "|$P(D_1 = 1 \\mid H)$|            1 |         0.01 |\n",
    "|$P(D_1 = 0 \\mid H)$|            0 |         0.99 |\n",
    ":label:`conditional_prob_D1`\n",
    "\n",
    "请注意，每列的加和都是1（但每行的加和不是），因为条件概率需要总和为1，就像概率一样。\n",
    "让我们计算如果测试出来呈阳性，患者感染HIV的概率，即$P(H = 1 \\mid D_1 = 1)$。\n",
    "显然，这将取决于疾病有多常见，因为它会影响错误警报的数量。\n",
    "假设人口总体是相当健康的，例如，$P(H=1) = 0.0015$。\n",
    "为了应用贝叶斯定理，我们需要运用边际化和乘法法则来确定\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1) \\\\\n",
    "=& P(D_1=1, H=0) + P(D_1=1, H=1)  \\\\\n",
    "=& P(D_1=1 \\mid H=0) P(H=0) + P(D_1=1 \\mid H=1) P(H=1) \\\\\n",
    "=& 0.011485.\n",
    "\\end{aligned}\n",
    "$$\n",
    "因此，我们得到\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(H = 1 \\mid D_1 = 1)\\\\ =& \\frac{P(D_1=1 \\mid H=1) P(H=1)}{P(D_1=1)} \\\\ =& 0.1306 \\end{aligned}.$$\n",
    "\n",
    "换句话说，尽管使用了非常准确的测试，患者实际上患有艾滋病的几率只有13.06%。\n",
    "正如我们所看到的，概率可能是违反直觉的。\n",
    "\n",
    "患者在收到这样可怕的消息后应该怎么办？\n",
    "很可能，患者会要求医生进行另一次测试来确定病情。\n",
    "第二个测试具有不同的特性，它不如第一个测试那么精确，\n",
    "如 :numref:`conditional_prob_D2`所示。\n",
    "\n",
    ":条件概率为$P(D_2 \\mid H)$\n",
    "\n",
    "| 条件概率 | $H=1$ | $H=0$ |\n",
    "|---|---|---|\n",
    "|$P(D_2 = 1 \\mid H)$|            0.98 |         0.03 |\n",
    "|$P(D_2 = 0 \\mid H)$|            0.02 |         0.97 |\n",
    ":label:`conditional_prob_D2`\n",
    "\n",
    "不幸的是，第二次测试也显示阳性。让我们通过假设条件独立性来计算出应用Bayes定理的必要概率：\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1, D_2 = 1 \\mid H = 0) \\\\\n",
    "=& P(D_1 = 1 \\mid H = 0) P(D_2 = 1 \\mid H = 0)  \\\\\n",
    "=& 0.0003,\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1, D_2 = 1 \\mid H = 1) \\\\\n",
    "=& P(D_1 = 1 \\mid H = 1) P(D_2 = 1 \\mid H = 1)  \\\\\n",
    "=& 0.98.\n",
    "\\end{aligned}\n",
    "$$\n",
    "现在我们可以应用边际化和乘法规则：\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(D_1 = 1, D_2 = 1) \\\\\n",
    "=& P(D_1 = 1, D_2 = 1, H = 0) + P(D_1 = 1, D_2 = 1, H = 1)  \\\\\n",
    "=& P(D_1 = 1, D_2 = 1 \\mid H = 0)P(H=0) + P(D_1 = 1, D_2 = 1 \\mid H = 1)P(H=1)\\\\\n",
    "=& 0.00176955.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "最后，鉴于存在两次阳性检测，患者患有艾滋病的概率为\n",
    "\n",
    "$$\\begin{aligned}\n",
    "&P(H = 1 \\mid D_1 = 1, D_2 = 1)\\\\\n",
    "=& \\frac{P(D_1 = 1, D_2 = 1 \\mid H=1) P(H=1)}{P(D_1 = 1, D_2 = 1)} \\\\\n",
    "=& 0.8307.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "也就是说，第二次测试使我们能够对患病的情况获得更高的信心。\n",
    "尽管第二次检验比第一次检验的准确性要低得多，但它仍然显著提高我们的预测概率。\n",
    "\n",
    "## 期望和方差\n",
    "\n",
    "为了概括概率分布的关键特征，我们需要一些测量方法。\n",
    "一个随机变量$X$的*期望*（expectation，或平均值（average））表示为\n",
    "\n",
    "$$E[X] = \\sum_{x} x P(X = x).$$\n",
    "\n",
    "当函数$f(x)$的输入是从分布$P$中抽取的随机变量时，$f(x)$的期望值为\n",
    "\n",
    "$$E_{x \\sim P}[f(x)] = \\sum_x f(x) P(x).$$\n",
    "\n",
    "在许多情况下，我们希望衡量随机变量$X$与其期望值的偏置。这可以通过方差来量化\n",
    "\n",
    "$$\\mathrm{Var}[X] = E\\left[(X - E[X])^2\\right] =\n",
    "E[X^2] - E[X]^2.$$\n",
    "\n",
    "方差的平方根被称为*标准差*（standard deviation）。\n",
    "随机变量函数的方差衡量的是：当从该随机变量分布中采样不同值$x$时，\n",
    "函数值偏离该函数的期望的程度：\n",
    "\n",
    "$$\\mathrm{Var}[f(x)] = E\\left[\\left(f(x) - E[f(x)]\\right)^2\\right].$$\n",
    "\n",
    "## 小结\n",
    "\n",
    "* 我们可以从概率分布中采样。\n",
    "* 我们可以使用联合分布、条件分布、Bayes定理、边缘化和独立性假设来分析多个随机变量。\n",
    "* 期望和方差为概率分布的关键特征的概括提供了实用的度量形式。\n",
    "\n",
    "## 练习\n",
    "\n",
    "1. 进行$m=500$组实验，每组抽取$n=10$个样本。改变$m$和$n$，观察和分析实验结果。\n",
    "2. 给定两个概率为$P(\\mathcal{A})$和$P(\\mathcal{B})$的事件，计算$P(\\mathcal{A} \\cup \\mathcal{B})$和$P(\\mathcal{A} \\cap \\mathcal{B})$的上限和下限。（提示：使用[友元图](https://en.wikipedia.org/wiki/Venn_diagram)来展示这些情况。)\n",
    "3. 假设我们有一系列随机变量，例如$A$、$B$和$C$，其中$B$只依赖于$A$，而$C$只依赖于$B$，能简化联合概率$P(A, B, C)$吗？（提示：这是一个[马尔可夫链](https://en.wikipedia.org/wiki/Markov_chain)。)\n",
    "4. 在 :numref:`subsec_probability_hiv_app`中，第一个测试更准确。为什么不运行第一个测试两次，而是同时运行第一个和第二个测试?\n"
   ]
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    "[Discussions](https://discuss.d2l.ai/t/1762)\n"
   ]
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